The Dollar-Euro Exchange Rate and Macroeconomic Fundamentals - A Time-varying Coefficient Approach
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The Dollar-Euro Exchange Rate and Macroeconomic
Fundamentals - A Time-varying Coefficient Approach
by
Joscha Beckmann*, Ansgar Belke** and Michael Kühl***
* University of Duisburg-Essen, ** University of Duisburg-Essen and IZA Bonn,
*** University of Goettingen
Essen, Berlin and Goettingen
September 2010
Abstract
This paper investigates the temporal stability of the relationship between the Deutschmark/
US dollar exchange rate and macroeconomic fundamentals. We use monthly data from
1975:01 to 2007:12. Applying a novel time-varying coefficient estimation approach, we come
up with some interesting properties of our empirical model. Firstly, there is no stable long-run
equilibrium relationship among fundamentals and exchange rates, since the breakdown of
Bretton Woods. Secondly, there are no recurring regimes, i.e. across different regimes, either
the coefficient values for the same fundamentals differ or the significance differs. Thirdly,
there is no regime into which no fundamentals enter. Fourthly, the deviations resulting from
the stepwise cointegrating relationship act as a significant error-correction mechanism. In
other words, we are able to show that fundamentals play an important role in determining the
exchange rate, but their impact differs significantly across different subperiods.
JEL codes: E44, F31, G12
Keywords: Structural exchange rate models, cointegration, structural breaks,
switching regression, time-varying coefficient approach
Corresponding author: Professor Dr. Ansgar Belke, University of Duisburg-Essen, Department of
Economics, Chair for Macroeconomics, D-45117 Essen, and IZA Bonn, e-mail: ansgar.belke@uni-
due.de, phone: + 49 201183 2277, fax: + 49 201 183 4181.
Acknowledgments: We are grateful for valuable comments from the participants of the Annual Meeting
of the German Economic Association (Verein für Socialpolitik), September 8-11, 2009, Magdeburg, of
the XII. Applied Economics Meeting Conference, held from June 4-6, 2009, Madrid, and of the 12th
Workshop on International Economic Relations, March 12-14, 2009, Goettingen. We gratefully
acknowledge from helpful comments from an anonymous referee.
11. Introduction
Disentangling the main drivers of exchange rates is still one of the most controversial
research areas in economics. After the first generation models of exchange rate
determination, which see the exchange rate as the relative price of domestic and foreign
monies (Dornbusch, 1976a,b; Frenkel, 1976; Kouri, 1976; Mussa, 1976) were brought to the
data, it became clear that exchange rate models can only partly be used to explain past
exchange rates with the help of fundamentals, and that they perform poorly in forecasting, in
particular (Meese and Rogoff, 1983 and 1988). The results of the seminal study by Meese
and Rogoff (1983) still represent the benchmark: exchange rate forecasts by structural
models can hardly outperform naïve random walk forecasts (Rogoff, 2009).
Since then, many contributions have tried to refute their results. Sticking to the implicit
assumption that exchange rates and fundamentals are cointegrated, and implementing
exogenous parameter restrictions, a couple of authors find predictability in the long run for a
similar period, as in Meese and Rogoff (Mark, 1995; Chinn and Meese, 1995). 1 However,
extending the estimation period yields mostly contrary findings (Kilian, 1999; Abhyankar et
al., 2005). A critical point is the implicit assumption of cointegration, which leads to biased
conclusions if a stable long-run relation does not exist (Berkowitz and Giorgianni, 2001).
While the empirical models of the late 1980s mostly neglected the potential existence
of a long-run relationship between the fundamentals and the exchange rate, structural
models were applied at the beginning of the 1990s which tested explicitly for a long-run
relationship among exchange rates and fundamentals. These kinds of empirical model,
which are based upon cointegration relationships, can indeed improve the evidence in favour
of predictability in the long run when periods up to the end of the 1990s are covered
(MacDonald and Taylor, 1993, 1994). 2 However, any extension of the sample period typically
yields a breakdown in cointegration relationships (Groen, 1999). Surprisingly, little attention
is directed to an examination of the link between exchange rates and fundamentals with
respect to structural changes in cases where cointegration does not hold.
Stock and Watson (1996) show that univariate and bivariate macroeconomic time
series are subject to substantial instabilities which result in poor forecasting performance.
Different market surveys suggest that various fundamentals are important during different
periods (Cheung and Chinn, 2001; Gehrig and Menkhoff, 2006). Bacchetta and Wincoop
(2009) argue that large and frequent variations in the relationship between the exchange rate
and macro fundamentals naturally develop when structural parameters in the economy are
1
Mark (1995) is the first author who focuses on more than one exchange rates simultaneously. He includes the
Canadian dollar, the Deutschmark, the Japanese yen and the Swiss franc expressed in US dollar. Chinn and
Meese (1995) do include the pound sterling in US dollars as well as the US dollar and the Deutschmark in
Japanese yen but not the Swiss franc.
2
MacDonald and Taylor (1994) investigate the pound sterling-US dollar exchange rate.
2unknown and subject to changes. As a consequence, market participants can give
“excessive” weight to some (macroeconomic) fundamentals during specific periods, i.e. to
so-called “scapegoats” (Bacchetta and van Wincoop, 2004). Parameter instabilities then
arise when the empirical realisation of such a scapegoat changes.
A similar explanation of parameter instabilities can be obtained from the imperfect
knowledge approach (e.g. Goldberg and Frydman, 1996b, 2007). This approach is based on
the view that market participants do not know the exact model but use fundamentals for
forecasting exchange rates in a way consistent with the assumed theory. Accordingly, the
link between fundamentals and the exchange rate changes when the market participants
revise their beliefs in the underlying model. Hence, it is reasonable to assume that a strong
and significant relationship between exchange rates and fundamentals exists during some
subperiods and that its nature tends to change considerably over time.
Goldberg and Frydman (1996a,b, 2001) report evidence that fundamentals do matter
in a way which is not entirely consistent with the monetary model during some subperiods of
floating while such evidence cannot be found during other periods. 3 Thus, the instability of
the monetary model in the data-generating process might serve as an explanation for the
findings of Cheung et al. (2005). The latter suggest that model specifications which work well
in one period do not necessarily work well in another period. 4 From this point of view, a
fundamental value of the exchange rate exists in the sense that a part of the exchange rate
movements is driven by fundamentals.
In the recent past, models capable of taking different regimes into account have been
applied to the monetary approach. 5 For instance, Sarno et al. (2004) use a Markov regime-
switching model in order to investigate the response of exchange rates to deviations from
fundamental values in different regimes. Sarno and Valente (2009) demonstrate that
exchange rate models that optimally use the information in the fundamentals often change,
which in turn implies frequent shifts in the coefficients. What is more, de Grauwe and
Vansteenkiste (2007) investigate particularly the adjustment of the nominal exchange with
respect to changes in the fundamentals under different inflation regimes. Taylor and Peel
3
The inability to find such evidence in other subperiods, such as the transition periods, does not mean that
fundamentals do not matter. Rather, this may be due to small sample sizes or specification error. Also, even in
the subperiods for which fundamentals are found to matter, the results are not entirely consistent with the
monetary models.
4
See also Bacchetta and Wincoop (2009). Parameter instability, i.e. an unstable relationship between exchange
rates and macro fundamentals, is confirmed by formal econometric evidence delivered by Rossi (2006).
5
For an analogous application to inflation and unemployment in the context of different political regimes see
Belke (2000). He interprets the significance of the error-correction parameter after regime-dependent structural
breaks in the long-run cointegrating relationship have been taken into account as empirical evidence of
hysteresis.
3(2000), Taylor et al. (2001) and Kilian and Taylor (2003) make use of models that allow for a
smooth transition between two states, supporting the hypothesis that exchange rate
adjustments towards equilibrium paths are nonlinear. To be more specific, fundamentals
become important if the deviation from an equilibrium rate is large.
Frömmel et al. (2005a,b) test directly for the significance of different regimes in the
exchange rate determination equation of the real interest rate differential model. However,
since the authors specify their model in first differences, they do not investigate a long-run
relationship in a strict sense. 6 Joining Goldberg and Frydman (1996a,b, 2001), the
coefficients in the exchange rate determination process itself are allowed to change within
their framework. All other contributions focus on deviations of the exchange rate from a
fundamental value which assumes cointegration with implied restrictions without modelling
the long-run structure separately.
Both of the above-mentioned regime-switching approaches, however, have in
common that they only allow for a fixed number of perseverative, i.e. regularly recurring,
regimes. In early works, Schinasi and Swamy (1989) and Wolff (1987) applied a time-varying
coefficient model (TVP) to monetary models. They were able to show that their models
displayed rather better forecasting properties than fixed coefficient models. Hence, the
consideration of time-varying coefficients appears to be a worthwhile next step towards a
valid empirical model of the exchange rate.
Taking these considerations as a starting point, we address several research
questions by using a general exchange rate determination model which is based upon the
monetary approach and nests a range of variants of the latter. Our working hypothesis is that
a relationship between the exchange rate and fundamentals continuously holds, but that its
composition varies considerably over time. To test our hypotheses we proceed as follows:
firstly, we check whether the long-run equilibrium relationship among some fundamentals
and the US dollar exchange rate vis-à-vis the Deutschmark/euro since the breakdown of
Bretton Woods I has been subject to structural changes. Secondly, we test whether the
estimated relationships represent cointegrating relations. The latter is the case if the
hypothesis of non stationarity of the error term resulting from the stepwise relationship can
be rejected. As regards our third hypothesis, we check empirically whether fundamentals
matter for each regime identified by us. Fourthly, we then test whether the regimes are not
perseverative, which would imply that the empirical realisation of the estimated coefficients
for specific fundamentals and/or their significance differs across different regimes. Fifthly, we
focus on a test of rational expectations in the tradition of Goldberg (2000). Finally, we test
6
In order to obtain a long-run perspective, Frömmel et al. (2005a,b) make use of annual changes constructed
from a monthly data set.
4whether the exchange rate adjusts to disequilibria and investigate whether the adjustment
speed tends to be stable.
The remainder of this paper is organized as follows. Section two gives a short
overview of the array of fundamental models we consider later on, and motivates coefficient
instability from a theoretical perspective. In section three we describe our econometric
methodology and in section 4 present the empirical results. We start with the estimation of a
multiple structural change model, as developed by Bai and Perron (1998, 2003), which we
apply to the reduced form of structural exchange rate models. As a next step, we make use
of the estimated breakpoints to generate indicator functions, and, based on these, we
estimate the structural model in order to obtain estimates for the different regimes. To this
purpose, we apply the fully modified OLS estimator by Phillips and Hansen (1990), which is
generally claimed to be able to deal with nonstationary variables as regressors and
regressands. Finally, we construct an error-correction term from the estimated relationships
and regress the change of the exchange rate on this error-correction term, in order to
investigate whether the exchange rate adjusts to deviations from a fundamental equilibrium
relationship. Section 5 concludes.
2. Monetary models of the exchange rate
2.1 Theories
After the breakdown of Bretton Woods I, exchange rate models were developed which see
exchange rates as asset prices (Dornbusch, 1976a; Frenkel, 1976; Kouri, 1976). All models
of this kind have in common that they rely on a stable money demand function of the form
M (1)
L(Y r , i)
P
with M representing the money supply, P the price level and L the money demand
depending on real income (Y) and interest rates (i). A basic assumption of the standard
monetary model is that the purchasing power parity (PPP) holds. In the log-linearized form,
the exchange rate can be expressed as the difference in price levels which is equal to the
difference between domestic and foreign money supply less real money demand based on
money market equations, so that the exchange rate is determined as follows:
s ( 1m y i) ( f
mf f
yf f
if ) (2)
2 3 1 2 3
f
1 m 1 mf 2 y 2
f
yf 3 i 3
f
if .
5In the literature, this model is widely known as the Frenkel and Bilson (FB) model.7 A
rise of the exchange rate s corresponds to depreciation of the domestic currency. In the
f
original monetary model is zero and 1 1
1 due to the structure of the money
demand function. Equation (2) can be rewritten under the restriction that the (semi-)
elasticities of the interest rates are equal. This yields:
s m f
mf y f
yf (i i f ). (3)
1 1 2 2 3
If the uncovered interest rate parity (UIP) holds, (i i f ) can be replaced by the
expected change in the exchange rate ( Et (st 1 ) st ) . With an expectation-generating
mechanism based upon PPP, the differences in interest rates can then be replaced by the
differences in expected rates of inflation. 8 Since it is known that the exchange rate often
deviates from the PPP the adjustment towards the PPP value can be taken into account in
addition to the expectations concerning the expected rates of
f 9
inflation Et (st 1 st ) ( st s) t t . The real interest rate model (RID) by Frankel
(1979) arises if the expectation formation process is combined with the UIP and is solved for
the expected change in the exchange rate (equation (4)).
s m f
mf y f
yf (it it f ) ( f
). (4)
1 1 2 2 3 4 t t
The negative sign of the interest rate differential implies that an increase in the
differential is associated with an appreciation of the domestic currency. With the help of
equation (4) a similar process can be explained as in the overshooting case of Dornbusch
(1976a). In Dornbusch (1976a) the exchange rate is negatively correlated with the interest
rate differential but without feedback on inflation expectations, i.e. 4 is zero. Equation (4)
allows the exchange rate to deviate from PPP in the short run, i.e. it reacts negatively on
interest rates, but still positively on inflation rate expectations.
A weakness of the traditional monetary model is that the real exchange rate is
assumed to be constant in the long run. Since it is expected that the PPP holds for traded
goods rather than for a mixture of traded and non-traded goods, as implicitly assumed when
using the overall price index, the prices of traded goods can be taken into account
(Dornbusch, 1976b). If the overall price index, which is determined by the money market,
consists of prices of both traded and non-traded goods, and if the PPP is only valid for traded
7
The terms are elasticities and is a constant term. The variables m and y are the logarithms of money
supply and real income. The interest rates are expressed as percentage.
8
This expression is equivalent to a money demand function in which the expected rates of inflation enter as
opportunity costs.
9
The parameter denotes the adjustment speed towards the equilibrium value s . The parameter denotes
the expected rate of inflation.
6goods, then the monetary approach yields an exchange rate determination equation in the
form:
f PtT Pt fT 10
s 1m 1 mf 2y 2
f
yf 3 (it it f ) 4( t t
f
) 6 6
f
. (5)
Pt NT Pt fNT
In the flex-price model, 4 is equal to zero and the exchange rate reacts positively to the
interest rate differential (Wolff, 1987). The proportion of traded to non-traded goods mirrors
the real exchange rate. A rise in the price of tradables relative to that of non-tradables
causes the nominal exchange rate to increase because the domestic good is substituted by
the foreign good. Such a rise might result from productivity differentials between countries as
expressed by the Harrod--Balassa--Samuelson effect (Harrod, 1939; Balassa 1964;
Samuelson 1964). Wu and Hu (2009) recently emphasized the importance of the Harrod--
Balassa--Samuelson effect when modelling deviations from purchasing power parity using an
ESTAR model.
In order to take account of real shocks, Hooper and Morton (1982) implement
changes of the equilibrium real exchange rate into the traditional monetary model (HM
model). In addition to nominal impact factors, the real side of the economy was introduced by
taking into consideration innovations in the current account. Hooper and Morton (1982) also
use overall trade balances as an indicator of the risk premium which arises from government
debt, an insufficient holding of international reserve, and foreign indebtedness. A fall in the
net foreign asset position (in particular if it is negative) raises the risk premium and, hence,
depreciates the domestic exchange rate. Hence, the risk premium reacts sensitively to a
worsening negative net foreign asset position. Thus, equation (4) can be extended by the
cumulated trade balances as a proxy for the overall trade balance (eq. (6)).11
s m f
mf y f
yf (it it f ) ( f
) CTBt f
CTB f . (6)
1 1 2 2 3 4 t t 5 5
In applied monetary models, equation (2) is typically estimated based by means of a
reduced form for which it is assumed that the elasticities of an economic variable are
f f f
identical in both countries. Hence, the restrictions 1 1 , 2 2 and 3 3 apply
(Meese and Rogoff, 1983). However, any analysis in which the coefficients are restricted to
be equal for each variable typically tends to result in biased coefficients (Haynes and Stone,
1981). If the structure of the economy is not known a priori, restricted coefficients do not help
in explaining the exchange rate. While the traditional monetary model assumes that domestic
and foreign assets are perfect substitutes, the assumption is relaxed by highlighting the role
of risk, as Hooper and Morton point out (1982). One model which explicitly takes risk premia
10
The parameter T denotes tradables and NT denotes non-tradables.
11
Since data on the current account are not available at a monthly frequency, it appears adequate to proxy the
current account by the trade balance.
7into account is the portfolio balance model (Branson, 1977). If a risk premium gains in
importance, it is preferable to use this portfolio balance approach. In such a case the
symmetry restriction regarding interest rates is relaxed because the domestic and foreign
bonds are not perfect substitutes. Using arguments stemming from the imperfect knowledge
approach, Goldberg (2000) has shown that a rejection of the symmetry restriction relating to
the interest rate differential is either linked to imperfect capital mobility or provides evidence
in favour of the imperfect knowledge approach over rational expectations. Although a precise
distinction between both explanations is not empirically possible, he concludes that the
absence of capital controls in most countries points towards the inadequacy of the rational
expectation hypothesis. (Goldberg, 2000).
In the following, we employ a hybrid model which picks up effects that can be found in
both monetary and portfolio models (Frankel, 1983). As a consequence, we remove the
restrictions of parameter equality in the interest rate differential and the inflation rate
differential in our equations (4) and (6). Thus, we start our analysis in as unrestrictive as
possible manner, bearing in mind the dynamics stemming from both the portfolio balance
approach and the monetary approach.
2.2 Long-run analysis with time-varying coefficients
Wolff (1987) gives three reasons why a time-varying coefficient model should be superior to
fixed-coefficient models. First of all, the money demand function is subject to instabilities,
which cause the coefficients in the exchange rate determination equation of a reduced model
to change (Leventakis, 1987). Another reason is given by the famous Lucas critique:
coefficients change if an anticipated change in the policy regime occurs. The third argument
is related to the long-run real exchange rate. The monetary model assumes that purchasing
power parity holds in the long run, from which follows that the long-run real exchange rate is
stable. Innovations in the real exchange rate from the real side of the economy can lead to
changes in the coefficients. Because we explicitly account for changes in the real exchange
rate, the latter issue deserves less attention in our analysis with respect to the choice of
estimation technique.
A reason for choosing time-varying coefficient models can also be derived from
different theories. In inter-temporal new open economy macroeconomic (NOEM) models
(Obstfeld and Rogoff, 1995), money demand does not depend on income, but on real
consumption. If we proxy real consumption by real income, a change in the average rate of
consumption results in a change in the elasticity of income in the exchange rate equation.
Thus, if consumption shares do vary, which is, for instance, true for the US, the exchange
rate determination equation thus also becomes time varying.
8As argued by Wilson (1979), an anticipated policy change, i.e. an expansionary
monetary policy, can generate a kind of dynamics which is different from those stemming
from unanticipated changes. Following Wilson (1979), the overshooting dynamics is slightly
different from those of Dornbusch (1976a). A very important result is that an appreciation
period of the domestic currency coincides with the increase in money supply, while in the
Dornbusch model a boost in money supply coincides with a depreciation of the former. If
anticipated and unanticipated shocks alternate, fixed coefficient models are inadequate
because they cannot capture both effects simultaneously. This argument is particularly
relevant if the frequency of observation is a monthly one. In such a case, these effects will
influence the long-run relationship and not enter the short-term dynamic.
Furthermore, the consistent expectations theory developed by Goldberg and Frydman
(1996a, 2001, 2007), which is based upon the imperfect knowledge approach, offers a broad
theoretical framework that is able to explain why some fundamentals might matter during
some time periods, but not during others. The authors argue that combinations of different
fundamentals need not be systematically similar, as market participants intermittently revise
their views as to how fundamentals influence the exchange rate. They show that
macroeconomic fundamentals can drive exchange rate swings. Such swings can therefore
be explained with the help of the basic relationships in a monetary model with either flexible
or sticky prices, if the assumption of rational expectations is replaced with an Imperfect
Knowledge representation of forecasting behaviour (Goldberg and Frydman, 2007). Within
this framework, market participants only have a rough knowledge concerning the link
between exchange rate and fundamentals, suggesting that they are only able to determine
the sign of the fundamentals with respect to their influence on the exchange rate. The
authors conclude that it is not reasonable to base an empirical analysis on a fully
predetermined model, as it is not possible to pre-specify either the fundamentals or the way
these fundamentals influence the exchange rate.
According to the results gained by Sarno et al. (2004), and de Grauwe and
Vansteenkiste (2007), the adjustment of exchange rates towards the long-run equilibrium
relationship also does not appear to be time-invariant. Consequently, we expect that
adjustment differs from period to period, at least over a long span of data. An adjustment
towards the long-run equilibrium relationship can occur because the exchange rate
predominantly reacts to the fundamentals, or because, conversely, the fundamentals react to
changes in exchange rates. In the latter case, it is possible that the exchange rate does not
adjust in subperiods. The changing of an adjustment coefficient can be due to the revision of
beliefs concerning the importance of macroeconomic factors. An increase should coincide
with a homogeneity of beliefs regarding the fundamental model. If no fundamental factor
9matters, the adjustment coefficient will be zero in the corresponding period. Consequently,
the adjustment coefficient has the potential to differ between subperiods.
Siklos and Granger (1997) have developed a framework which appears to be well
suited to analyzing these issues in the necessary detail. They point out that a cointegration
relationship can be subject to structural changes, and argue that the common stochastic
trends are only present in specific periods. In this respect, they introduce the concept of
regime-sensitive cointegration, or “switch on – switch off” cointegration. In addition to a time-
varying cointegration vector, their framework also allows the causality between the variables
to change during the period of observation. This means that the dimension of the vector
which contains the adjustment coefficients can be reduced during subperiods.
In our long-run relationship analysis we are thus potentially simultaneously confronted
with switch on and off cointegration, a changing cointegration vector and the adjustment
process. The main difficulty inherent in our estimations, then, is coping with potential
overlaps of these phenomena. Hence, our approach takes account of different regimes. It is
able to distinguish between cases in which the cointegration relationship is switched on and
those in which different adjustments are present. In this paper, our working hypothesis is that
cointegration is continuously present over the whole period of observation, while only the
composition of the cointegration vector changes. An empirical rejection of this hypothesis,
which can be observed if either no fundamental factor enters the cointegration relationship or
the exchange rate does not adjust to disequilibria from the estimated long-run relationship, is
compatible with the results of Goldberg and Frydman (1996a, 2001, 2007) who inspired our
approach quite heavily. Our approach in principle delivers the same empirical pattern as their
setting: different fundamentals matter in different ways during different time periods and the
resulting regimes are not perseverative. Nevertheless, some differences remain. Whereas
our aim is to show that cointegration is continuously present, with only the composition of the
vector changing, the study of Goldberg and Frydman (196a,b, 2001) in principle allows for
the possibility that cointegration does not exist during subperiods. For example, the results of
Goldberg and Frydman’s (1996b) structural change analysis imply a couple of subperiods
that are too small to estimate a relationship between exchange rates and fundamentals.
However, this does not necessarily imply that fundamentals do not matter during these
subperiods.
For a multivariate case we consider the term
Yt t β t Xt t (7)
with
Xt [ X t1 ,..., X tk ] for n 1,..., K , (8)
10where K represents the maximum number of explanatory variables. 12 The matrix X t has the
dimension K 1 and β t the dimension 1 K . In our empirical analysis, we put the
following composite model under closer scrutiny:
'
f f f f pT p fT f
Yt st , X t m yi m y i CTB CTB (9)
p NT p fNT .
This model nests all models described in section 2.1. Consequently, we can use this
equation to assess the empirical validity of the presented models in section 4 by applying
Wald tests.
3. Modeling structural changes and estimating cointegrating relations -
methodological issues
3.1 Testing for multiple structural changes
In general, two frameworks for tests for structural change can be distinguished. The first one
consists of generalized fluctuation tests in which a model is fitted to the data and an empirical
process is derived that captures these fluctuations either in the residuals or in coefficient
estimates. If the generated process exceeds the boundaries of the limiting process, which
can be derived from the functional central limiting theorem, the null hypothesis of parameter
constancy has to be rejected. This implies that a structural change occurs at the
corresponding point in time (Zeileis et al., 2003).
The classical and the OLS based CUSUM test and the fluctuation test of Nyblom
(1989) are well-known examples of such kind of methods. These structural change tests are
predominantly designed for stationary variables. In the case of a cointegration analysis an
eigenvalue fluctuation test developed by Hansen and Johansen (1999) which heavily relies
upon Nyblom can be applied. While these procedures have the advantage of not assuming a
particular pattern of deviation from the null hypothesis they can either only identify a single
break or show general instability.
The second framework to test for structural changes is to compare the OLS residuals
from regressions for different subsamples. This can be done, for example, by applying the F-
statistics or the Chow test. In this paper, we exclusively adopt an extension of the latter case
developed by Bai and Perron (1998, 2003). Their basic idea is to choose breakpoints such
that the sum of squared residuals for all observations is minimized.
12
The term t denotes a regime-dependent constant term. The variable t represents an error term.
11As a starting point, consider a multiple linear regression with m breakpoints and m+1
regimes
yt xt' κ z t' δ j ut , (t Tj 1 1,..., T j ) , (10)
for j 1,..., m 1 with the convention that T0 0 and Tm 1 T . The term y t denotes the
dependent variable, xt and z t denominate the regressors and κ and δ are the coefficient
vectors. Note that only δ varies over time while κ is constant.
With a sample of T the first step is to calculate the corresponding values for all
possible T T 1 2 segments.13 The estimated breakpoints T1 ...... Tm by definition represent
the linear combination of these segments which achieve a minimum of the sum of squared
residuals (Bai and Perron, 2003). Formally:
(Tˆ1 ,..., Tˆm ) arg min T1 ,...,Tm ST (T1 ,..., Tm ). (11)
Bai and Perron (2003) develop a dynamic programming algorithm which compares all
possible combinations of the segments. Their methodology allows testing for multiple
structural breaks under different conditions. 14 Within our framework, the location of the
breakpoints is also obtained by calculating the sum of squared residuals. To select the
dimension of the model we apply the Bayesian Information Criterium (BIC) which according
to Bai and Perron (2003) works well in most cases when breaks are present. After calculating
the tests for all possible breakpoints the sequence Tˆ1 ,..., Tˆm is selected as the configuration
at which the BIC achieves its minimum. Carrion-i-Silvestre and Sansó (2006) show that this
approach yields a consistent estimate of the break fraction. The breakpoints obtained in this
fashion are a local minimum of the sum of squared residuals given the number of
breakpoints but not necessary a global minimum.
It is important to note that the procedure of Bai and Perron has originally been
developed for the case of stationary variables (I(0)). Nevertheless, it can as well be applied
13
Bai and Perron (1998) note that for practical purposes less than T (T 1) segments are permissible, for
example if a minimum distance between each break is imposed. In the framework of this paper, breaks are
allowed to occur every 12 months.
14
One possibility is to test the null of no change against the hypothesis of a fixed number of breaks m k using
F- tests based on the sum of squared residuals under both hypotheses. For an unknown number of breaks, one
way is to allow a maximum number of breaks. In this case one can apply the so called double maximum test. The
number of breakpoints is then selected by comparing the F-values described above for the different numbers of
breakpoints and select the configuration with the highest F-value respectively the minimum of the sum of the
squared residuals. Another possibility is to test sequentially for an additional break using the “l vs. l+1” break
tests. For details see Bai and Perron (1998, 2003).
12to nonstationary variables which are integrated of order one (I(1)). For instance, Siklos and
Granger (1997) use this methodology to identify structural breaks in the interest parity
equation between the United States and Canada in the context of regime-sensitive
cointegration. In addition, Zumaquero and Urrea (2002) point out that the break estimator is
consistent also in the nonstationary case. Using disaggregated price indexes for seven
countries, they test for structural breaks in the coefficients of cointegrating relations which
represent absolute and relative purchasing power parity. They also examine instabilities in
the adjustment behaviour of price ratios and exchange rates. Finally, Kejriwal and Perron
(2008) demonstrate that the results of Bai and Perron (1998) in general continue to hold
even with I(0) and I(1) variables in the regression. 15 This is also true if one allows for
endogenous I(1) regressors. 16 The use of information criteria as the BIC is also correct in
both cases.
To check our results for robustness, we also apply the CUSUM test combined with
Andrews and Ploberg (1996) in a similar way as Goldberg and Frydman (2001) to detect
possible breakpoints. However, with no considerable differences arising from the results, we
proceed using the breakpoints obtained by the Bai and Perron methodology.
3.2 Estimating cointegrating relations with single equations
After identifying the breakpoints we now turn to the issue of correct estimation. As Bai and
Perron’s methodology is designed for single equations, we cannot consider multivariate
system estimators as proposed by Johansen (1988) or Stock and Watson (1988). Besides
the traditional approach of Engle and Granger (1987), several modified single estimators
have been developed. Examples are the fully modified estimator by Phillips and Hansen
(1990) and the approach of Engle and Yoo (1991). 17 Even in the case of a multi-dimensional
cointegration space, single equation approaches can be used to achieve asymptotically
efficient estimates of single cointegrating relationships.
For our purposes, the fully modified (FM) estimator is the most suitable method. In
contrast to traditional single equation formulas it considers endogenous regressors (Phillips,
1991). Phillips and Hansen (1990) show that the FM-OLS estimator is hyperconsistent for a
unit root in single equations autoregression. Phillips (1995) proves that this procedure is
reliable in the case of full rank or cointegrated I(1) regressors 18 as well as with I(0)
regressors. Hargreaves (1994) runs a Monte Carlo simulation and points out that single
15
This is only true if, as in our case, the intercept is allowed to change across segments.
16
For the case without unit roots, Perron and Yamamoto (2008) show that the estimation of the break dates via
OLS is preferable to an IV procedure in the presence of endogenous regressors.
17
For a review of the different estimation methods of estimating cointegrating relationships see Hargreaves
(1994), Phillips and Loretan (1991) and Caporale and Pittis (1999).
18
Note that the direction of cointegration does not need to be known. Regressors containing a deterministic trend
are also allowed.
13estimators, in general, are robust if more than one cointegrating relation exists, with the FM-
OLS estimator doing best. He concludes that the FM-OLS estimator should be preferred,
even in advance of multivariate methods, if one wants to examine one cointegrating vector
and is unsure about the cointegrating dimensionality. This is of particular interest for this
paper, as we are primarily interested in the long-run relationship between exchange rates
and fundamentals, and do not wish to pay too much attention to other cointegrating
relationships which might arise between the reported fundamentals. Caporale and Pittis
(1999) claim that the FM-OLS estimator and the Johansen estimator perform best in finite
samples.19 Goldberg and Frydman (2007) use the systems approach developed by Phillips
(1991), which is similar to the FM-OLS method for testing for cointegration between the
exchange rate and fundamentals in a regime-sensitive framework.
The root idea of this concept is to estimate cointegrating relations directly by
correcting traditional OLS with regard to endogeneity and serial correlation (Phillips, 1995).
Let z t denominate an n -vector where y t denotes an r -dimensional I(1) process while X t
is an (n r ) ((n r )1 (n r ) 2 -dimensional vector of cointegrated or possibly stationary
regressors. u t represents an n-vector stationary time series. Both vectors can be partitioned
as follows:
yt u1t
zt x1t ut u 2t .
x 2t u 3t (12)
The data generating process of yt is represented by the following cointegrated relation
yt βx 1t u1t .
(13)
The vectors of the regressors are specified as follows
Δx1t u 2t , (14)
x 2t u3t . (15)
The estimator corrections can be applied without pre-testing the regressors for unit
roots as both corrections can be conducted by treating all components of x t as
nonstationary. For the nonstationary components, this transformation reduces asymptotically
to the ideal correction while the differenced stationary components vanish asymptotically.
Such a correction does not have any effect on the subvectors of x t where serial correlation
19
Furthermore, also Phillips and Hansen (1990), Hargreaves (1994) and Cappucio and Lubian (2001) report good
finite sample properties of the FM-OLS estimator.
14or endogeneity are not present. 20 A further advantage is that we do not have to account for
cointegration between the x 1t regressors within this methodology (Phillips, 1995).
To imply the corrections, we first consider the long-run covariance matrix which
can be decomposed into a contemporaneous variance and the sums of auto-covariances
(Hargreaves, 1994).
E (u t u t' ) k 2
E (u 0 u k' ) k 2
E (u k u 0' ) (16)
'
(17)
We define as
. (18)
Estimation of these covariance parameters can be achieved by using the pre-
whitened kernel estimator suggested by Andrews and Monahan (1992). 21 The endogeneity
correction then has the form
y t* yt ˆ ˆ 1ΔX .
0x xx t (19)
The above correction is employed to account for endogeneities in the regressors x 0 t
linked with any cointegration between x 0 t and y t . The second correction takes into account
the effects of serial covariances in the shocks u1t and any serial covariance between u 0 t and
the history of u1t . The bias effect arises from the persistence of shocks due to the unit roots
in x1t . The induced one-sided long-run covariance matrices carry these effects in an OLS
regression (Phillips, 1995). They can be defined as
ˆ ˆ ˆ ˆ 1 ˆ .
0x 00 0x xx x0 (20)
20
Without serial correlation or endogeneity the FM-OLS estimator is identical to the OLS estimator.
21
Other studies adopt the estimator of Newey and West (1987) which is robust to serial correlati on and
heteroskedasticity. For details see Cappuccio and Lubian (2001).
15The correction is then given by
ˆ* ˆ ˆ ˆ 1 ˆ .
0x 0x 0x xx xx (21)
Combining both corrections the formula for the fully modified estimator is 22
*
ˆ* (Y *' X T )( X ' X ) 1 . (22)
0x
3.3 Regime shifts in cointegration models
To apply the FM-OLS estimator in a model with structural changes we proceed in a similar
way as Hansen (2003) does in the Johansen framework by allowing the coefficients to
change their values at the breakpoints.23
We rewrite equation ( 22 ) with (t ) as a constant
yt (t ) xt' (t ) ut . (23)
The piecewise constant time-varying coefficients are given by
j (t ) 0 1
1 1t ... 1 ,
m mt (24)
j (t ) 1
1 1t ... m mt 1 (25)
where the indicator function for each subsample is defined as follows (Hansen, 2003)
1mt 1(T j 1 1 t Tj ) J with j 1,..., m (26)
with the convention that T0 0 and Tm T . Defining dummies according to the indicator
function ensures that we are able to obtain estimates for each period.
In a similar way, the error correction representation can be rewritten by allowing for structural
changes in the adjustment process.
yt (t ) (t ) y t 1 (t 1) x 't 1 (t 1) et (27)
(t ) in eq. (27) is a constant and et the residuals from the error correction model. The
term (t ) represents the adjustment coefficient concerning deviations from the long-run
22 ˆ Y ' X ( X ' X ) 1.
The traditional OLS estimator is given by β
23
We corroborated our results with a related approach introduced by Gregory and Hansen (1996). They model
the changes in the intercept and the slope coefficients relative to the first subperiod as a benchmark, running from
0 to T1 . The base model is then written as y t 1 (t ) x 't κ 1 x 't κ (t ) z t' δ1 z t' δ j (t ) u t .
16equilibrium. Similarly to eq. (25), a corresponding indicator function can be defined for (t ) .
The indicator function for (t ) is equivalently equal to eq. (24).
4. Data and estimated models
4.1 Data
Our sample contains monthly data running from January 1975 until December 2007. We use
the aggregate M1 for money supply. Real income is proxied by the real production index. As
suggested by Wolff (1987) the producer price index serves as a proxy for tradable goods
while the basket of non-tradables is reflected by the consumer price index (CPI).
Furthermore, we use the overall trade balance as an approximation of the current account.
As seen in the Hooper--Morton model, the equilibrium flow determines the equilibrium stock.
For the short-term interest rates we use money market rates with a maturity of three months.
Exchange rates, money supply and real income are expressed in logarithms. All series are
seasonally adjusted and are taken from International Financial Statistics of the International
Monetary Fund.
In strong contrast to other studies investigating the euro exchange rate, we rely on
the Deutschmark and the fundamentals of Germany before the introduction of the euro. The
reason is that we are interested in market rates which could be contrasted by using weighted
ECU-Data. In a sense, the Deutschmark has been a predecessor of the euro as it had a
similar importance on the foreign exchange market. One reason was the big influence of the
German Bundesbank (Fratianni and von Hagen, 1990). We therefore use a time series which
contains the German values until December 1998 and, from then on, the values of the euro
area. Consequently, the Deutschmark / US dollar exchange rate is converted by the official
Deutschmark / euro exchange rate in order to obtain a level adjustment. As a consequence,
we also adjust the German fundamentals in levels to allow for a smooth transition to the euro
area data. Since we deal with structural break models in the empirical section, we do not see
any problems with our proceeding. The reason is that if a break due to data adjustment were
important, the Bai--Perron test would signify a break around January 1999.
4.2 Preliminary tests for unit roots and stationarity
Although the FM-OLS estimator and the Bai--Perron methodology are basically able
to handle a combination of I(0) and I(1) regressors, testing the data for unit roots is
necessary as a first step. With the exchange rate being an I(1) variable, the concept of
cointegration only makes sense if the fundamentals can also be treated as I(1) processes. By
definition, a cointegrating relationship can only exist between variables which are integrated
of the same order (Engle and Granger, 1987). Neither can a stationary variable force a
nonstationary variable to adjust, nor is a stationary relationship between I(1) and I(2)
17variables possible. The distinction between the I(1) and I(2) variables is important in our
context as there is much evidence in the literature that it is better to treat macroeconomic
time series, like money supplies and exchange rates, as I(2) rather than I(1) processes. In
those cases, a standard I(1) analysis might lead to biased conclusions (Juselius, 2006). 24
To test for unit roots, we apply the Phillips--Perron (PP), the Kwiatkowski--Philips--
Schmidt--Shin (KPSS) and the GLS-based Dickey--Fuller (DF-GLS) tests. In the first
instance, we test for stationarity in the levels. Differences are taken and tested again if a unit
root remains, i.e. if the corresponding variables are integrated of order two. If both
hypotheses are rejected we conclude that the variable is I(2). According to our results, most
of our variables can be considered as being integrated of order one. The results of the tests
are presented in Table 1.
Table 1 about here
However, in a few cases, the evidence is mixed. For instance, our results for the
cumulated overall trade balance suggest that this variable is integrated of order two.25
Therefore, we decide to work with first differences of the US and the euro area trade balance
series. This can be done without changing the underlying economic theory. What is more,the
KPSS test rejects the null hypothesis of stationarity of the change in the US money supply
and the second difference of the trade balance of the euro area. However, since the other
tests indicate I(1) properties of the respective series we treat them all as I(1).
4.3 Empirical results
4.3.1 Assessing the stability of the long-run relationship
We now derive the main hypotheses, to be tested in the following, from the arguments
developed in section 2. Our first hypothesis concerns the stability of a long-run exchange
rate determination equation and runs as follows:
H1: There is no stable long-run relationship between the fundamentals and the
EUR/USD exchange rate.
If the empirical application of the Bai--Perron test corroborates the existence of
structural breaks, we cannot reject the validity of hypothesis H1. We present the breakpoints
identified by applying the Bai--Perron methodology in Table 2. With an eye on the fact that
24
Frydman et al. (2010) account for this issue by using an I(2) framework to analyze long swings in the
Deutschmark / US dollar exchange rate.
25
The results are available on request.
18we are able to identify eight breakpoints, we feel legitimized to state that breaks occur quite
frequently. Hence, we cannot reject H1 and conclude that a stable long-run relationship
among the variables does not exist.
Table 2 about here
An important question is whether some of these breakpoints are related to major
economic or political events. The first two breakpoints located in July of 1977 and September
1981 cannot be matched up with one specific incident, although the second date refers to the
so-called pseudo-monetarism policy of the Federal Reserve of 1979 and 1982 (Timberlake,
1993). The instability during the mid-1980s coincides with the end of the rise of the US dollar.
During that time it had been officially stated by the authorities that the strong dollar was no
longer wanted, as it harmed the US economy (Destler and Henning, 1989).
The next breakpoint, located around October 1988 (row 4, Table 2), might be traced
back to a specific monetary policy stance. In 1988, the monetary policy stance on both sides
of the Atlantic, i.e. that of the US Fed and the Bundesbank, became more restrictive. Besides
the usual monetary policy suspects, the election of George Bush Senior and the G-7 summit
in Berlin26 offer further and quite popular explanations.
Whereas any meaningful interpretation of the breakpoint of 1991 appears to be quite
arbitrary, the assessment of the following instability in 1993 appears to be more
straightforward. It is usually attributed to the crisis of the European Monetary System.
Significant changes in the US and German monetary policies at this time are also taken into
account by many scholars.
After a relatively stable period up to the end of the 1990s, the next instability emerges
shortly after the start of EMU. The last break in 2004 coincides exactly with an event which
saw the short-term interest rates of the euro area declining below the level of US interest
rates. Of course, as far as the dating of breakpoints and their economic interpretation are
concerned, we prefer to follow quite standard appraisals. Nor should one forget that many
other important developments are not reflected by breakpoints. Furthermore, it remains a
difficult task to identify the exact trigger which caused the observed instabilities.
Nevertheless, it seems that policy announcements seem to play an important role in
determining the breakpoints detected by the Bai and Perron procedure. We leave a closer
examination of the identified breakpoints to future research.
26
In contrast to previous meetings, the participants of the Berlin meeting did not publically claim that fluctuations
in the dollar were unwanted.
194.3.2 Testing for cointegration between the exchange rate and fundamentals
Our second hypothesis is related to the question of whether the estimated relationship can
actually be interpreted as a cointegration relationship. The corresponding hypothesis runs as
follows:
H2: The estimated relationship can be interpreted as a cointegrating relationship
between exchange rates and fundamentals.
H2 can be investigated by applying unit root tests to the error term. If we are able to
reject the null of non-stationarity according to the unit root test results, we feel legitimized to
conclude that H2 holds. As a first step, we estimate equation (23) by FM-OLS, using the
obtained break dates displayed in Table 2. The corresponding empirical results are
presented in Table 3. They will be analyzed in more detail in section 4.3.3.
Table 3 about here
In order to check whether the relationship obtained from the FM-OLS estimation can
truly be interpreted as a cointegration relationship, we apply unit root tests to the resulting
error series, strictly following the idea of residual-based cointegration tests. In doing so, we
have to apply critical values which take account of the number of estimated coefficients.
Because of the huge number of coefficients used in our estimation we should not rely on the
standard critical values provided by the literature. For this reason, we separately run a
Monte-Carlo simulation with 10,000 repetitions in order to obtain critical values for our
model.27 According to the results of the DF-GLS and the PP test reported in Table 4 the error
term resulting from our step-wise relationship should be considered as stationary. This in turn
conveys clear evidence in favour of a long-run cointegrating relationship between the
exchange rate and its fundamentals. Hence, we accept our second hypothesis H2.
Table 4 about here
4.3.3 Estimation and interpretation of the long-run relationship
We proceed by putting the results of the FM-OLS estimation under closer scrutiny. The
validation of H2 raises the question whether the exchange rate is linked to fundamental
27
To be more precise, we construct the data generating process for each variable. Each process is constructed
as an independent random walk. In addition, we take account for the breaks obtained by each model.
Consequently, the null hypothesis is no cointegration, meaning that we obtain a series for the error term that
contains a unit root for each model. The critical values can then be drawn from the realized distribution. However,
this methodology cannot be applied to the KPSS test which assumes stationarity under the null. In this case , we
would need to know the exact specification of the cointegration relationship under the consideration of our breaks
to obtain relevant critical values. We therefore decided to leave out the KPSS test and to rely on the DF-GLS and
the PP Test.
20factors during each regime. In order to check this, our third hypothesis runs as follows:
H3: There is no regime in the step-wise long-run relationship in which no fundamental
factor enters.
One option to assess the validity of hypothesis H3 is to apply Wald tests to our
composite model which we estimate by means of FM-OLS. Under the null hypothesis, all
coefficients except the constant terms are restricted to zero. Any empirical rejection of this
null hypothesis confirms our hypothesis H3. The results concerning these restrictions can be
found in column 1 of Table 5.
Table 5 about here
This hypothesis is clearly rejected at the 1% level in all cases, implying that at least
one coefficient except the intercept term is different from zero. Hence, we feel legitimized to
argue that H3 is corroborated, i.e. that at least one fundamental variable is significant with
respect to the exchange rate (as a non-rejection would have implied that no fundamentals
matter).
Since we accept H3, the next interesting question is whether some of the regimes are
perseverative. As already mentioned in section 1, many studies assume that the relationship
between exchange rate and fundamentals can be described by models that distinguish
between two perseverative regimes. Hence, we move on to our fourth hypothesis:
H4: There is no perseverative regime in the step-wise long-run relationship.
As a prerequisite of our test of this hypothesis, we implement restrictions aimed at
achieving the structure of the theoretical models outlined in section 2.1 for our estimated
composite model. In order to test the validity of the RID model, we restrict step by step the
coefficients of money supply, income, inflation and both interest rates to zero. The results
can be seen in columns 3 to 6 of Table 5. A rejection of the null hypothesis in principle yields
evidence in favour of the RID model. As a next step, we restrict only the two relative prices to
zero. A rejection of this hypothesis yields the importance of the purchasing power parity
based upon prices of tradables. In the same vein, a rejection of the hypothesis that the
coefficients of the cumulated current account are zero delivers evidence that these factors
are important.
Our strategy for checking the validity of hypothesis H4 starts from these Wald tests.
First, we assess empirically whether there are similar combinations concerning the rejection
or non-rejection of the null hypotheses regarding the subsequent Wald tests. If there is no
similar combination, H4 is already confirmed; if there are similar combinations we additionally
21inspect our estimated regimes. H4 can then not be rejected if at least one coefficient is
significant in one regime, while this is not the case in the other regime(s). What is more, the
models can also only be confirmed if the signs of the estimated coefficients are in line with
underlying theory. Thus, we have to look at the sign of the estimated coefficient in the
corresponding regime in order to verify general consistency with a model.
The results of the different tests presented in Table 5 clearly suggest that the
variables included in the RID are significant and, hence, important. Altogether, we find similar
results only for the periods starting from 1985 and 1999, as the null hypothesis is always
rejected in both cases. However, comparing the results of these periods with respect to the
estimation results of Table 4, many coefficients are significant in one period but not in
another. Thus, the suspected linkage between exchange rates and fundamentals differs in
each period. Hence, we can confirm H4.
As a next step, we take the results for the different regimes displayed in Table 3
under closer scrutiny, with regard to the consistency of the different model configurations. An
interesting result is that in cases of significance both inflation rates always enter the equation
with the correct sign. The same is true in most cases for the estimated coefficients of the US
money supply and the US tradable to non-tradable price ratio, while in many cases the
corresponding German and European coefficient signs are not consistent with theory.
Overall, our results are broadly consistent with the real interest rate model (Equation 4) in the
first two subperiods, after our period of observation has started (row 1 and 2 of Table 3).
From this point of view, our empirical results clearly corroborate the findings in the literature
concerning the early period after the breakdown of Bretton Woods I.28 The significant
coefficients for the period from 1991 to 1993 always enter with the correct signs.
Furthermore, the tradable-non-tradable price ratio of the United States is the only significant
variable that enters with the wrong sign during the last period. In all other cases the pattern
of the estimation results is less clear, as some coefficients enter with signs that are not
consistent with standard theory while others do reflect theoretical considerations. However,
some fundamentals gain in significance in each period. Thus, we can conclude that the
relationship between exchange rates and fundamentals over a period of at least one and a
half years is stable (otherwise the Bai--Perron test would have estimated more breaks, as
our configuration allows for breaks every 12 months). However, it is not possible to confirm
one specific model over the whole period, as the signs and the significance levels of the
coefficients differ across the periods. Although fundamentals seem to matter, the standard
exchange rate models considered in this paper do not provide a complete explanation of how
they do. Another interesting finding is that the US-variables seem to enter more often with
28
For an early overview see, for example, Isard (1987).
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